As pointed out by Schreiber [1]}, there are many reasons why \(X_{t}\)  and \(Y_{t+\tau }\)  might share information.
Firstly, both \(X\)  and \(Y\)  may be synchronized, and so \(X_{t}\)  predicts \(Y_{t+\tau }\)  in the same fashion that \(Y_{t}\)  would, and so it would be disingenuous to attribute that shared information to information flow.
Similarly, \(X\)  and \(Y\)  may be jointly influenced by a third system \(Z\) , and so there is no direct, or even indirect, information flow from \(X\)  to \(Y\) .
Schreiber referred to these two situations as the two time series being correlated via common history and common input signals, and proposed discounting these influences from the time-delayed mutual information via conditioning:
\(\mathcal {I}(X_{t} ; Y_{t + \tau } | Y_{t})=\sum _{x_t,y_t,y_{t+\tau }}p(y_{t+\tau },y_{t},x_t)\log _2 \left(\frac{p(y_{t+\tau }|y_t,x_t)}{p(y_{t+\tau }|y_t)}\right)~,\) 
